For any two complex numbers z₁ and z₂, prove that

Re (z₁ z₂) = Re z₁ Re z₂ – Imz₁ IMz₂

It is given in the question that,

_z1 and z₂ are two complex numbers and we have to prove that:

_{Re (z1}z₂) = Rez₁ Rez₂ – Imz₁ Imz₂

_{For this, firstly let z1} = x₁ + iy₁ and z₂ = x₂ + iy₂

_{Thus, z1}z₂ = (x₁ + iy₁) (x₂ + iy₂)

_{= x1} (x₂ + iy₂) + iy₁ (x₂ + iy₂)

_{= x1}x₂ + ix₂y₂ + iy₁x₂ + i²y₁y₂

_{= x1}x₂ + ix₂y₂ + iy₁x₂ – y₁y₂ (i² = - 1)

_{= (x1}x₂ – y₁y₂) + i (x₁y₂ + y₁x₂)

_{= Re (z1}z₂) = x₁x₂ – y₁y₂

_{∴ Re (z1}z₂) = Rez₁ Rez₂ – Imz₁ Imz₂

_{Hence, proved}